| | | |

## Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers

#### Madihah KHALİD [1] , Zulmaryan EMBONG [2]

This paper presents the findings of an intervention study that emphasizes concrete representation to improve students’ conceptual understanding in learning mathematics. The study specifically examined the effectiveness of focusing on concrete representations – the algebra tiles, in minimizing students’ errors in the operation on integers. A quasi-experimental design with a sample of 60 students from two intermediate Year 7 classes was employed in this study. The control group and the experimental group consisted of 30 students each, chosen through purposive sampling. Data from pre and post-tests and field notes were collected and analysed to measure changes from the intervention. The quantitative data from the tests showed an increase from 14.70 to 23.47 for the experimental group, as compared to 18.67 to 22.57 for the control group. The ANCOVA returns statistically significant results that can be attributed to the intervention strategies. Data from field notes indicate students’ improvement in problem-solving skills, and students’ interest in the lessons and motivation to learn. This study suggests that teaching with emphasis on concrete representation improves students’ conceptual understanding. Hence students’ understanding of integers was enhanced due to the promotion of concepts through manipulatives, pictures, verbal and symbolic representation which were also employed during the intervention. This study may be useful to teachers who usually encounter problems when teaching this topic.
Algebraic tiles, Teaching and Learning integers, intervention, multiple representation, manipulative
• Ahn, S., & Choi, J. (2014, April). A synthesis of the quantitative literature on students’ mathematics achievement. Paper presented at the American Educational Research Association, San Diego, CA.
• Badarudin, B. R., & Khalid, M. (2008). Using the Jar Model to Improve Students’ Understanding of Operations on Integers. In B. Gomez., D. De Bock, & Z. Usiskin (Eds), Proceedings of ICME-11–topic study group 10 research and development in the teaching and learning of number systems and arithmetic (Vol. 1, pp. 85-94) ICME Mexico.
• Badarudin, B. R., & Khalid, M. (2009). Investigating students’ common errors in integers. In: Boorer, D., Perera Q., Wooed, K. (Eds.) Evolving pedagogies: Meeting the global challenges of diversity and interdependence (pp. 233 – 250). Brunei: Universiti Brunei Darussalam Press.
• Bny Rosmah, B. (2006). Investigating Teaching and Learning of Integers at Form 1 level in Brunei Darussalam. (Unpublished Master thesis). University of Brunei Darussalam, Brunei.
• Boaler, J. (2015). What's math got to do with it?: How teachers and parents can transform mathematics learning and inspire success. Penguin Books.
• Brahier, D. J. (2016). Teaching secondary and middle school mathematics. New York: Routledge
• Creswell, J. W., & Creswell, J. D., (2017). Design Approach: Qualitative, quantitative and mixed-methods approaches (5th ed.). Singapore: Sage Publication.
• Embong, Z. (2020). Analysing pupils’ errors in operations of integers among form 1 pupil. (Unpublished Doctora dissertation). IIUM, Malaysia.
• Fuadiah, N. F., Suryadi, D. & Turmudi. (2017). Some difficulties in understanding negative numbers faced by students: A qualitative study applied at secondary schools in Indonesia. International Education Studies; 10(1), 24-38. https://doi.org/10.5539/ies.v10n1p24
• Hayes, B. & Stacey, K. (1999, July). Teaching negative number using integer tiles. In J. M. Truran & K. M. Truran (Eds.), Making the difference In Proceedings of the 22nd annual conference of the mathematics education research group of Australasia (MERGA, p.573), Adelaide, Australia.
• Khalid, M., & Embong, Z. (2020). Sources and possible causes of errors and misconceptions in operations of integers. International Electronic Journal of Mathematics Education, 15(2), em0568.
• Khalid, M., Ibrahim, M. B., Saad, S., Othman, J., Mohd Yussuf, Y., Embong, Z., (2018, November). A preliminary study on form 1 students errors and misconception in operations of integers. In R. Embong, H.M. Lateh (Eds) Proceeding of National Education Deans’ Council Seminar (pp. 115 – 124), UNISZA, Malaysia.
• Khalid, M (2017). Fostering problem solving and performance assessment among Malaysian mathematics teachers. Sains Humanika, 9, (1-2).
• Korn, J. (2014). Teaching conceptual understanding of mathematics via a hands-on approach. (Unpublished Senior Honours Thesis). Liberty University, USA.
• Kuchemann, D. E. (1981) Algebra.In K.M. Hart (Ed) Children’s understanding of mathematics: 11-16 (pp. 102 – 119). London: John Murray.
• Laerd Statistics (2020). One way ANCOVA in SPSS statistics. Retrieved from https://statistics.laerd.com/spss-tutorials/ancova-using-spss-statistics.php
• Leitze, A. R., & Kitt, N. A. (2000). Using homemade algebra tiles to develop algebra and pre-algebra concepts. Mathematics Teacher, 93(6), 462-520.
• Lesh, R. (1979). Mathematical learning disabilities: considerations for identification, diagnosis and remediation. In R. Lesh, D. Mierkiewicz, & M.G. Kantowski (Eds), Applied mathematical problem solving (pp. 235-264). Ohio: ERIC/SMEAC.
• Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. Teaching and Learning of Mathematics, 21, 33-40.
• Makonye, J. P., & Fakude, J. (2016). A study of errors and misconceptions in the learning of addition and subtraction of directed numbers in grade 8. SAGE Open, 6(4), 2158244016671375.
• Meyer, D. (2018, October). “What does fluency without understanding look like?” Retrieved from https://blog.mrmeyer.com/2018/what-does-fluency-without-understanding-look-like/
• Moses, B. E. (1977). The nature of spatial ability and its relationship to mathematical problem solving. (Unpublished doctoral dissertation). Indiana University: USA. National Assessment of Educational Progress [NAEP]. (2019). 2019 Mathematics Assessment, Institute of Education Sciences, National Center for Education Statistics, U.S. Department of Education.
• National Research Council. (2001). Adding it up: Helping children learn mathematics. In Kilpatrick, J. Swafford, & B. Findell (Eds.). Washington, DC: National Academy Press.
• Okpube, N. M. (2016). Card games and algebra tic tacmatics on achievement of junior Secondary II students in algebraic expressions. International Journal of Evaluation and Research in Education, 5(2), 93-100. DOI: http://doi.org/10.11591/ijere.v5i2.4527
• Osman, S., Che Yang, C. N. A., Abu, M. S., Ismail, N., Jambari, H., & Kumar, J. A. (2018). Enhancing students’ mathematical problem-solving skills through bar model visualisation technique. International Electronic Journal of Mathematics Education, 13(3), 273-279. https://doi.org/10.12973/iejme/3919
• Sadler, J. T. (2012). The positives about negatives: A study of errors and misconceptions with integer operations in adult education (Unpublished master’s thesis). State University of New York.
• Saraswati, S. (2016). Supporting students’ understanding of linear equations with one variable using algebra tiles. Journal on Mathematics Education, 7(1), 21-32. DOI: 10.22342/jme.7.1.2814.19-30
• Schindler, M., & Hubmann, S. (2013). About Student’s individual concepts of negative integer: in terms of the order relation. In B. Ubuz, C. Hacer, & M.A. Mariotti, (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (pp. 373–382). Ankara, Turkey: ERME and METU.
• Schlosser, T. K. (2010). Using algebra tiles to aid students in factoring polynomials (Unpublished master’s thesis). Central Connecticut State University, New Britain.
• Sharp, J. M. (1995, October). Results of using algebra tiles as meaningful representations of algebra concepts. Paper presented at the Annual Meeting of the Mid-Western Education Research Association, Chicago.
• Steiner, C. J. (2009). A study of pre-service elementary teachers’ conceptual understanding of integers (Unpublished doctoral dissertation). Kent State University College and Graduate School of Education, US.
• Stokes, S. (2002). Visual literacy in teaching and learning: a literature perspective. Electronic Journal for the Integration of Technology in Education, 1(1), 10-19.
• Thornton, S. (2001). A picture is worth a thousand words. Retrieved February 15, 2017 from math.unipa.it/~grim/AThornton251.PDF
• Toh, T.S., Tengah, K.A., Shahrill, M., Tan, A. & Leong, E. (2017, April). The flipped classroom strategy: The effects of implementation at the elementary school level mathematics lessons. In Proceeding of the 3rd International Conference on Education 2017 (ICEDU 2017), (pp. 186-197) Kuala Lumpur, Malaysia.
• Widjaja, W., Stacey, K., & Steinle, V. (2011). Locating negative decimals on the number line: Insights into the thinking of pre-service primary teachers. Journal of Mathematical Behavior, 30(1), 80-91. https://doi.org/10.1016/j.mathb.2010.11.004.
Primary Language en Education and Educational Research Research Articles Author: Madihah KHALİD (Primary Author)Institution: International Islamic University MalaysiaCountry: Malaysia Orcid: 0000-0002-5653-1631Author: Zulmaryan EMBONGInstitution: International Islamic University MalaysiaCountry: Malaysia Ministry of Education, Malaysia [FRGS/1/2016/SSI09/UIAM/02/10] This research was supported by [FRGS/1/2016/SSI09/UIAM/02/10]. We would like to thank the Ministry of Education for providing us with the grant and make this research possible. Publication Date : December 15, 2020
 Bibtex @research article { turkbilmat775605, journal = {Turkish Journal of Computer and Mathematics Education (TURCOMAT)}, issn = {}, eissn = {1309-4653}, address = {}, publisher = {Türkbilmat Eğitim Hizmetleri}, year = {2020}, volume = {11}, pages = {762 - 773}, doi = {10.16949/turkbilmat.775605}, title = {Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers}, key = {cite}, author = {Khali̇d, Madihah and Embong, Zulmaryan} } APA Khali̇d, M , Embong, Z . (2020). Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers . Turkish Journal of Computer and Mathematics Education (TURCOMAT) , 11 (3) , 762-773 . DOI: 10.16949/turkbilmat.775605 MLA Khali̇d, M , Embong, Z . "Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers" . Turkish Journal of Computer and Mathematics Education (TURCOMAT) 11 (2020 ): 762-773 Chicago Khali̇d, M , Embong, Z . "Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers". Turkish Journal of Computer and Mathematics Education (TURCOMAT) 11 (2020 ): 762-773 RIS TY - JOUR T1 - Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers AU - Madihah Khali̇d , Zulmaryan Embong Y1 - 2020 PY - 2020 N1 - doi: 10.16949/turkbilmat.775605 DO - 10.16949/turkbilmat.775605 T2 - Turkish Journal of Computer and Mathematics Education (TURCOMAT) JF - Journal JO - JOR SP - 762 EP - 773 VL - 11 IS - 3 SN - -1309-4653 M3 - doi: 10.16949/turkbilmat.775605 UR - https://doi.org/10.16949/turkbilmat.775605 Y2 - 2020 ER - EndNote %0 Türk Bilgisayar ve Matematik Eğitimi Dergisi Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers %A Madihah Khali̇d , Zulmaryan Embong %T Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers %D 2020 %J Turkish Journal of Computer and Mathematics Education (TURCOMAT) %P -1309-4653 %V 11 %N 3 %R doi: 10.16949/turkbilmat.775605 %U 10.16949/turkbilmat.775605 ISNAD Khali̇d, Madihah , Embong, Zulmaryan . "Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers". Turkish Journal of Computer and Mathematics Education (TURCOMAT) 11 / 3 (December 2020): 762-773 . https://doi.org/10.16949/turkbilmat.775605 AMA Khali̇d M , Embong Z . Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers. Turkish Journal of Computer and Mathematics Education (TURCOMAT). 2020; 11(3): 762-773. Vancouver Khali̇d M , Embong Z . Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers. Turkish Journal of Computer and Mathematics Education (TURCOMAT). 2020; 11(3): 762-773. IEEE M. Khali̇d and Z. Embong , "Emphasizing Concrete Representation to Enhance Students’ Conceptual Understanding of Operations on Integers", Turkish Journal of Computer and Mathematics Education (TURCOMAT), vol. 11, no. 3, pp. 762-773, Dec. 2020, doi:10.16949/turkbilmat.775605

Authors of the Article
[2]